This question is basically about comparison of IC sheaves (or their sheaf cohomologies) for the settings: 1. variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear, 2. variety is over a finite field $F_q$ and sheaf is $\overline{\mathbb{Q}}_l$-linear. ($l$ and $q$ coprime)
I am trying to understand the proof of Kazhdan-Lusztig conjecture. Let $X$ be the flag variety for a semisimple algebraic group $G$, where both of them are defined over $\mathbb{C}$. Let $W$ be the Weyl group. After using Beilinson-Bernstein localization theorem and Riemann-Hilbert correspondence, the proof of Kazhdan-Lusztig conjecture boils down to a computation of Euler-characteristic of stalk cohomology of IC sheaves. More precisely, let $X_{w}$ be the Schubert cell for $w\in W$ and $IC_{X_w}$ be the IC complex for the Schubert variety $\overline{X_w}$. The proof reduces to showing $$P_{y,w}(1)=\sum_i (-1)^i \text{dim} H^i_{yB}(IC_{X_w}) $$ for $y,w\in W$. Note that we are working on setting 1; variety is over $\mathbb{C}$ and sheaf is $\mathbb{C}$-linear.
In the paper Representations of Coxeter groups and Hecke algebras by Kazhdan-Lusztig, they compute the ($q^{1/2}$-graded) Euler characteristic for stalk cohomology. Let the flag variety $X$ and the group $G$ defined over a finite field $F_q$. Consider l-adic IC sheaves $IC_{X_w}$. Then the paper shows that $$ P_{y,w}(q)=\sum_i (-1)^i q^{i/2} \text{dim} H^i_{yB}(IC_{X_w}). $$ Note that this theorem is considered in setting 2; variety is over a finite field $F_q$ and sheaf is $\overline{\mathbb{Q}}_l$-linear. Here comes my question:
are we able to deduce the first theorem from the second theorem?
I saw in some references the proof of the first theorem, but the arguments did not use the second theorem. They used mixed Hodge modules (D-Modules, Perverse Sheaves, and Representation Theory) or semisimple complexes (Perverse Sheaves and Applications to Representation Theory).
I have thought about three possible scenarios. Let $X$ be a variety and $Y$ be its smooth locally closed subvariety and that both of them can be defined over $\mathbb{C}$ and $F_q$.
- There is a comparison theorem between stalk cohomologies of IC sheaves $IC_Y$ in two settings.
- Note that the Frobenius action on stalk cohomology $H^i_{yB}$ of IC sheaves in setting 2 is proved to be $q^{i/2}$, therefore the second theorem proves that the graded trace of Frobenius action is $P_{y,w}(q)$. Moreover, Euler characteristic is a graded trace of trivial action. So it suffices to show the following: for $X$ and $Y$ as in the assumption, we can compare graded trace of Frobenius action on stalks of $IC_Y$ in the setting 2 with graded trace of trivial action (which we imagine to be the `0-th power of Frobenius action') on stalks of $IC_Y$ in the setting 1.
- On setting 1, we can use small resolution $\pi: \hat{X}_w\rightarrow X_w$ of Schubert varieties. Then computing stalk cohomology at the point $yB$ is the same as computing singular cohomology of fiber $\pi^{-1}(yB)$. If we can do a similar thing in setting 2, then the problem boils down to comparison between singular cohomology of the variety $\pi^{-1}(yB)$ and $l$-adic etale cohomology of the same variety defined over $F_q$.
